nLab
separated geometric morphism

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Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Definition

A geometric morphism f:𝒳→𝒴 of toposes is separated if the diagonal 𝒳→𝒳× 𝒴𝒳 is a proper geometric morphism.

In particular if 𝒴 is the terminal object in Topos, hence the canonical base topos Set, we say that a topos 𝒳 is a Hausdorff topos if 𝒳→𝒳×𝒳 is a proper geometric morphism.

More generally, since there is a hierarchy of notions of proper geometric morphism, there is accordingly a hierarchy of separatedness conditions.

Examples

Proposition

For G a discrete group and BG=(G→→*) its delooping groupoid, the presheaf topos GSet≃[BG,Set] is Hausdorff precisely if G is a finite group.

In (Johnstone) this is example C3.2.24

References

Chapter II of

  • Ieke Moerdijk, Jacob Vermeulen, Relative compactness conditions for toposes (pdf) and Proper maps of toposes , American Mathematical Society (2000)

Around def. C3.2.12 of

Revised on May 9, 2012 03:54:31 by Zoran Škoda (31.45.129.110)
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